Special examples of Robert Bryants answer are Lie groups.
On any Lie group, the left trivialization of the tangent bundle corresponds to a flat connection whose torsion is essentially the Lie bracket, whereas the right trivialization corresponds to another connection without curvature whose torsion is essentially the negative of the Lie bracket. 

On the other hand any inner product on the Lie algebra gives rise to a right invariant metric (biinvariant if the inner product is invariant under the adjoint representation).
Its Levi-Civita connections is torsion free and it has a very interesting curvature
(See a paper by Milnor for the finite dimensional case or the famous paper by Arnold on volume preserving diffeomorphisms).